Prev

Next

Index-> contents reference index search external

Up-> CppAD library OdeGear

CppAD-> Install Introduction AD ADFun multi_thread library cppad_ipopt_nlp Example preprocessor Appendix

library-> ErrorHandler NearEqual speed_test SpeedTest time_test NumericType CheckNumericType SimpleVector CheckSimpleVector nan pow_int Poly LuDetAndSolve RombergOne RombergMul Runge45 Rosen34 OdeErrControl OdeGear OdeGearControl BenderQuad opt_val_hes LuRatio CppAD_vector thread_alloc memory_leak

OdeGear-> OdeGear.cpp

Headings-> Syntax Purpose Include Fun ---..t ---..x ---..f ---..f_x ---..Warning m n T X e Scalar Vector Example Source Code Theory Gear's Method

An Arbitrary Order Gear Method
Syntax
# include <cppad/ode_gear.hpp>
OdeGear(F, m, n, T, X, e)

Purpose
This routine applies Gear's Method to solve an explicit set of ordinary differential equations. We are given $\mathit{f}:\mathbf{R}\times {\mathbf{R}}^{\mathit{n}}\to {\mathbf{R}}^{\mathit{n}}$ be a smooth function. This routine solves the following initial value problem $$\begin{array}{ccc}\hfill \mathit{x}({\mathit{t}}_{\mathit{m}-1})& \hfill =\hfill & {\mathit{x}}^0\hfill \\ \hfill {\mathit{x}}^{\prime }(\mathit{t})& \hfill =\hfill & \mathit{f}[\mathit{t},\mathit{x}(\mathit{t})]\hfill \end{array}$$ for the value of $\mathit{x}({\mathit{t}}_{\mathit{m}})$ . If your set of ordinary differential equations are not stiff an explicit method may be better (perhaps Runge45 .)

Include
The file cppad/ode_gear.hpp is included by cppad/cppad.hpp but it can also be included separately with out the rest of the CppAD routines.

Fun
The class Fun and the object F satisfy the prototype
     Fun &F
This must support the following set of calls
     F.Ode(t, x, f)
     F.Ode_dep(t, x, f_x)

t
The argument t has prototype
     const Scalar &t
(see description of Scalar below).

x
The argument x has prototype
     const Vector &x
and has size n (see description of Vector below).

f
The argument f to F.Ode has prototype
     Vector &f
On input and output, f is a vector of size n and the input values of the elements of f do not matter. On output, f is set equal to $\mathit{f}(\mathit{t},\mathit{x})$ (see f(t, x) in Purpose ).

f_x
The argument f_x has prototype
     Vector &f_x
On input and output, f_x is a vector of size $\mathit{n}*\mathit{n}$ and the input values of the elements of f_x do not matter. On output, $$\mathit{f}\_\mathit{x}[\mathit{i}*\mathit{n}+\mathit{j}]={\partial }_{\mathit{x}(\mathit{j})}{\mathit{f}}_{\mathit{i}}(\mathit{t},\mathit{x})$$
Warning
The arguments f, and f_x must have a call by reference in their prototypes; i.e., do not forget the & in the prototype for f and f_x.

m
The argument m has prototype
     size_t m
It specifies the order (highest power of $\mathit{t}$ ) used to represent the function $\mathit{x}(\mathit{t})$ in the multi-step method. Upon return from OdeGear, the i-th component of the polynomial is defined by $${\mathit{p}}_{\mathit{i}}({\mathit{t}}_{\mathit{j}})=\mathit{X}[\mathit{j}*\mathit{n}+\mathit{i}]$$ for $\mathit{j}=0,\dots ,\mathit{m}$ (where $0\le \mathit{i}<\mathit{n}$ ). The value of $\mathit{m}$ must be greater than or equal one.

n
The argument n has prototype
     size_t n
It specifies the range space dimension of the vector valued function $\mathit{x}(\mathit{t})$ .

T
The argument T has prototype
     const Vector &T
and size greater than or equal to $\mathit{m}+1$ . For $\mathit{j}=0,\dots \mathit{m}$ , $\mathit{T}[\mathit{j}]$ is the time corresponding to time corresponding to a previous point in the multi-step method. The value $\mathit{T}[\mathit{m}]$ is the time of the next point in the multi-step method. The array $\mathit{T}$ must be monotone increasing; i.e., $\mathit{T}[\mathit{j}]<\mathit{T}[\mathit{j}+1]$ . Above and below we often use the shorthand ${\mathit{t}}_{\mathit{j}}$ for $\mathit{T}[\mathit{j}]$ .

X
The argument X has the prototype
     Vector &X
and size greater than or equal to $(\mathit{m}+1)*\mathit{n}$ . On input to OdeGear, for $\mathit{j}=0,\dots ,\mathit{m}-1$ , and $\mathit{i}=0,\dots ,\mathit{n}-1$ $$\mathit{X}[\mathit{j}*\mathit{n}+\mathit{i}]={\mathit{x}}_{\mathit{i}}({\mathit{t}}_{\mathit{j}})$$ Upon return from OdeGear, for $\mathit{i}=0,\dots ,\mathit{n}-1$ $$\mathit{X}[\mathit{m}*\mathit{n}+\mathit{i}]\approx {\mathit{x}}_{\mathit{i}}({\mathit{t}}_{\mathit{m}})$$
e
The vector e is an approximate error bound for the result; i.e., $$\mathit{e}[\mathit{i}]\ge |\mathit{X}[\mathit{m}*\mathit{n}+\mathit{i}]-{\mathit{x}}_{\mathit{i}}({\mathit{t}}_{\mathit{m}})|$$ The order of this approximation is one less than the order of the solution; i.e., $$\mathit{e}=\mathit{O}({\mathit{h}}^{\mathit{m}})$$ where $\mathit{h}$ is the maximum of ${\mathit{t}}_{\mathit{j}+1}-{\mathit{t}}_{\mathit{j}}$ .

Scalar
The type Scalar must satisfy the conditions for a NumericType type. The routine CheckNumericType will generate an error message if this is not the case. In addition, the following operations must be defined for Scalar objects a and b:

Operation	Description
a < b	less than operator (returns a bool object)

Vector
The type Vector must be a SimpleVector class with elements of type Scalar . The routine CheckSimpleVector will generate an error message if this is not the case.

Example
The file OdeGear.cpp contains an example and test a test of using this routine. It returns true if it succeeds and false otherwise.

Source Code
The source code for this routine is in the file cppad/ode_gear.hpp.

Theory
For this discussion we use the shorthand ${\mathit{x}}_{\mathit{j}}$ for the value $\mathit{x}({\mathit{t}}_{\mathit{j}})\in {\mathbf{R}}^{\mathit{n}}$ which is not to be confused with ${\mathit{x}}_{\mathit{i}}(\mathit{t})\in \mathbf{R}$ in the notation above. The interpolating polynomial $\mathit{p}(\mathit{t})$ is given by $$\mathit{p}(\mathit{t})=\sum _{\mathit{j}=0}^{\mathit{m}}{\mathit{x}}_{\mathit{j}}\frac{\prod _{\mathit{i}\ne \mathit{j}}(\mathit{t}-{\mathit{t}}_{\mathit{i}})}{\prod _{\mathit{i}\ne \mathit{j}}({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{i}})}$$ The derivative ${\mathit{p}}^{\prime }(\mathit{t})$ is given by $${\mathit{p}}^{\prime }(\mathit{t})=\sum _{\mathit{j}=0}^{\mathit{m}}{\mathit{x}}_{\mathit{j}}\frac{\sum _{\mathit{i}\ne \mathit{j}}\prod _{\mathit{k}\ne \mathit{i},\mathit{j}}(\mathit{t}-{\mathit{t}}_{\mathit{k}})}{\prod _{\mathit{k}\ne \mathit{j}}({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{k}})}$$ Evaluating the derivative at the point ${\mathit{t}}_{\ell }$ we have $$\begin{array}{ccc}\hfill {\mathit{p}}^{\prime }({\mathit{t}}_{\ell })& \hfill =\hfill & {\mathit{x}}_{\ell }\frac{\sum _{\mathit{i}\ne \ell }\prod _{\mathit{k}\ne \mathit{i},\ell }({\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{k}})}{\prod _{\mathit{k}\ne \ell }({\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{k}})}+\sum _{\mathit{j}\ne \ell }{\mathit{x}}_{\mathit{j}}\frac{\sum _{\mathit{i}\ne \mathit{j}}\prod _{\mathit{k}\ne \mathit{i},\mathit{j}}({\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{k}})}{\prod _{\mathit{k}\ne \mathit{j}}({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{k}})}\hfill \\ \hfill & \hfill =\hfill & {\mathit{x}}_{\ell }\sum _{\mathit{i}\ne \ell }\frac{1}{{\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{i}}}+\sum _{\mathit{j}\ne \ell }{\mathit{x}}_{\mathit{j}}\frac{\prod _{\mathit{k}\ne \ell ,\mathit{j}}({\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{k}})}{\prod _{\mathit{k}\ne \mathit{j}}({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{k}})}\hfill \\ \hfill & \hfill =\hfill & {\mathit{x}}_{\ell }\sum _{\mathit{k}\ne \ell }({\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{k}}{)}^{-1}+\sum _{\mathit{j}\ne \ell }{\mathit{x}}_{\mathit{j}}({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\ell }{)}^{-1}\prod _{\mathit{k}\ne \ell ,\mathit{j}}({\mathit{t}}_{\ell }-{\mathit{t}}_{\mathit{k}})/({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{k}})\hfill \end{array}$$ We define the vector $\mathrm{\alpha }\in {\mathbf{R}}^{\mathit{m}+1}$ by $${\mathrm{\alpha }}_{\mathit{j}}=\{\begin{array}{cc}\sum _{\mathit{k}\ne \mathit{m}}({\mathit{t}}_{\mathit{m}}-{\mathit{t}}_{\mathit{k}}{)}^{-1}\hfill & \mathrm{if}\mathit{j}=\mathit{m}\hfill \\ ({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{m}}{)}^{-1}\prod _{\mathit{k}\ne \mathit{m},\mathit{j}}({\mathit{t}}_{\mathit{m}}-{\mathit{t}}_{\mathit{k}})/({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{k}})\hfill & \mathrm{otherwise}\hfill \end{array}$$ It follows that $${\mathit{p}}^{\prime }({\mathit{t}}_{\mathit{m}})={\mathrm{\alpha }}_0{\mathit{x}}_0+\cdots +{\mathrm{\alpha }}_{\mathit{m}}{\mathit{x}}_{\mathit{m}}$$ Gear's method determines ${\mathit{x}}_{\mathit{m}}$ by solving the following nonlinear equation $$\mathit{f}({\mathit{t}}_{\mathit{m}},{\mathit{x}}_{\mathit{m}})={\mathrm{\alpha }}_0{\mathit{x}}_0+\cdots +{\mathrm{\alpha }}_{\mathit{m}}{\mathit{x}}_{\mathit{m}}$$ Newton's method for solving this equation determines iterates, which we denote by ${\mathit{x}}_{\mathit{m}}^{\mathit{k}}$ , by solving the following affine approximation of the equation above $$\begin{array}{ccc}\hfill \mathit{f}({\mathit{t}}_{\mathit{m}},{\mathit{x}}_{\mathit{m}}^{\mathit{k}-1})+{\partial }_{\mathit{x}}\mathit{f}({\mathit{t}}_{\mathit{m}},{\mathit{x}}_{\mathit{m}}^{\mathit{k}-1})({\mathit{x}}_{\mathit{m}}^{\mathit{k}}-{\mathit{x}}_{\mathit{m}}^{\mathit{k}-1})& \hfill =\hfill & {\mathrm{\alpha }}_0{\mathit{x}}_0^{\mathit{k}}+{\mathrm{\alpha }}_1{\mathit{x}}_1+\cdots +{\mathrm{\alpha }}_{\mathit{m}}{\mathit{x}}_{\mathit{m}}\hfill \\ \hfill \left[{\mathrm{\alpha }}_{\mathit{m}}\mathit{I}-{\partial }_{\mathit{x}}\mathit{f}({\mathit{t}}_{\mathit{m}},{\mathit{x}}_{\mathit{m}}^{\mathit{k}-1})\right]{\mathit{x}}_{\mathit{m}}& \hfill =\hfill & \left[\mathit{f}({\mathit{t}}_{\mathit{m}},{\mathit{x}}_{\mathit{m}}^{\mathit{k}-1})-{\partial }_{\mathit{x}}\mathit{f}({\mathit{t}}_{\mathit{m}},{\mathit{x}}_{\mathit{m}}^{\mathit{k}-1}){\mathit{x}}_{\mathit{m}}^{\mathit{k}-1}-{\mathrm{\alpha }}_0{\mathit{x}}_0-\cdots -{\mathrm{\alpha }}_{\mathit{m}-1}{\mathit{x}}_{\mathit{m}-1}\right]\hfill \end{array}$$ In order to initialize Newton's method; i.e. choose ${\mathit{x}}_{\mathit{m}}^0$ we define the vector $\mathrm{\beta }\in {\mathbf{R}}^{\mathit{m}+1}$ by $${\mathrm{\beta }}_{\mathit{j}}=\{\begin{array}{cc}\sum _{\mathit{k}\ne \mathit{m}-1}({\mathit{t}}_{\mathit{m}-1}-{\mathit{t}}_{\mathit{k}}{)}^{-1}\hfill & \mathrm{if}\mathit{j}=\mathit{m}-1\hfill \\ ({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{m}-1}{)}^{-1}\prod _{\mathit{k}\ne \mathit{m}-1,\mathit{j}}({\mathit{t}}_{\mathit{m}-1}-{\mathit{t}}_{\mathit{k}})/({\mathit{t}}_{\mathit{j}}-{\mathit{t}}_{\mathit{k}})\hfill & \mathrm{otherwise}\hfill \end{array}$$ It follows that $${\mathit{p}}^{\prime }({\mathit{t}}_{\mathit{m}-1})={\mathrm{\beta }}_0{\mathit{x}}_0+\cdots +{\mathrm{\beta }}_{\mathit{m}}{\mathit{x}}_{\mathit{m}}$$ We solve the following approximation of the equation above to determine ${\mathit{x}}_{\mathit{m}}^0$ : $$\mathit{f}({\mathit{t}}_{\mathit{m}-1},{\mathit{x}}_{\mathit{m}-1})={\mathrm{\beta }}_0{\mathit{x}}_0+\cdots +{\mathrm{\beta }}_{\mathit{m}-1}{\mathit{x}}_{\mathit{m}-1}+{\mathrm{\beta }}_{\mathit{m}}{\mathit{x}}_{\mathit{m}}^0$$
Gear's Method
C. W. Gear, ``Simultaneous Numerical Solution of Differential-Algebraic Equations,'' IEEE Transactions on Circuit Theory, vol. 18, no. 1, pp. 89-95, Jan. 1971.

---

Input File: cppad/ode_gear.hpp